This considers function from $R^n$ into $R^m$ where one or both of $n$ or $m$ is greater than $1$.
functions $f:R \rightarrow R$ are called univariate functions.
functions $f:R^n \rightarrow R$ are called scalar functions
functions $\vec{r}:R\rightarrow R^m$ are parameterized curves. The trace of a parameterized curve is a path.
function $F:R^n \rightarrow R^m$, may be called vector fields in applications. They are also used to describe transformations.
When $m>1$ a function is called vector valued.
When $n>1$ the argument may be given in terms of components, e.g. $f(x,y,z)$; with a point as an argument, $F(p)$; or with a vector as an argument, $F(\vec{a})$. The identification of a point with a vector is done frequently.
Limits when $m > 1$ depend on the limits of each component existing.
Limits when $n > 1$ are more complicated. One characterization is a limit at a point $c$ exists if and only if for every continuous path going to $c$ the limit along the path for every component exists in the univariate sense.
The derivative of a univariate function, $f$, at a point $c$ is defined by a limit:
\[ ~ f'(c) = \lim_{h\rightarrow 0} \frac{f(c+h)-f(c)}{h}, ~ \]
and as a function by considering the mapping $c$ into $f'(c)$. A characterization is it is the value for which
\[ ~ |f(c+h) - f(h) - f'(c)h| = \mathcal{o}(|h|), ~ \]
meaning after dividing the left-hand side by $|h|$ the expression goes to $0$ as $|h|\rightarrow 0$. This characterization will generalize with the norm replacing the absolute value, as needed.
The derivative of a function $\vec{r}: R \rightarrow R^m$, $\vec{r}'(t)$, is found by taking the derivative of each component. (The function consisting of just one component is univariate.)
The derivative satisfies
\[ ~ \| \vec{r}(t+h) - \vec{r}(t) - \vec{r}'(t) h \| = \mathcal{o}(|h|). ~ \]
The derivative is tangent to the curve and indicates the direction of travel.
The tangent vector is the unit vector in the direction of $\vec{r}'(t)$:
\[ ~ \hat{T} = \frac{\vec{r}'(t)}{\|\vec{r}(t)\|}. ~ \]
The path is parameterized by arc length if $\|\vec{r}'(t)\| = 1$. In this case an "$s$" is used for the parameter, as a notational hint: $\hat{T} = d\vec{r}/ds$.
The Normal vector is the unit vector in the direction of the derivative of the tangent vector:
\[ ~ \hat{N} = \frac{\hat{T}'(t)}{\|\hat{T}'(t)\|}. ~ \]
In dimension $m=2$, if $\hat{T} = \langle a, b\rangle$ then $\hat{N} = \langle -b, a\rangle$ or $\langle b, -a\rangle$ and $\hat{N}'(t)$ is parallel to $\hat{T}$.
In dimension $m=3$, the binormal, $\hat{B}$, is the unit vector $\hat{T}\cross\hat{N}$.
The Frenet-Serret formulas define the curvature, $\kappa$, and the torsion, $\tau$, by
\[ ~ \begin{align} \frac{d\hat{T}}{ds} &= & \kappa \hat{N} &\\ \frac{d\hat{N}}{ds} &= -\kappa\hat{T} & & + \tau\hat{B}\\ \frac{d\hat{B}}{ds} &= & -\tau\hat{N}& \end{align} ~ \]
These formulas apply in dimension $m=2$ with $\hat{B}=\vec{0}$.
The curvature, $\kappa$, can be visualized by imagining a circle of radius $r=1/\kappa$ best approximating the path at a point. (A straight line would have a circle of infinite radius and curvature $0$.)
The chain rule says $(\vec{r}(g(t))' = \vec{r}'(g(t)) g'(t)$.
A scalar function, $f:R^n\rightarrow R$, $n > 1$ has a partial derivative defined. For $n=2$, these are:
\[ ~ \begin{align} \frac{\partial{f}}{\partial{x}}(x,y) &= \lim_{h\rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}\\ \frac{\partial{f}}{\partial{y}}(x,y) &= \lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}. \end{align} ~ \]
The generalization to $n>2$ is clear - the partial derivative in $x_i$ is the derivative of $f$ when the other$x_j$ are held constant.
This may be viewed as the derivative of the univariate function $(f\circ\vec{r})(t)$ where $\vec{r}(t) = p + t \hat{e}_i$, $\hat{e}_i$ being the unit vector of all $0$s except a $1$ in the $i$th component.
The gradient of $f$, when the limits exist, is the vector-valued function for $R^n$ to $R^n$:
\[ ~ \nabla{f} = \langle \frac{\partial{f}}{\partial{x_1}}, \frac{\partial{f}}{\partial{x_2}}, \dots \frac{\partial{f}}{\partial{x_n}} \rangle. ~ \]
The gradient satisfies:
\[ ~ \|f(\vec{x}+\Delta{\vec{x}}) - f(\vec{x}) - \nabla{f}\cdot\Delta{\vec{x}}\| = \mathcal{o}(\|\Delta{\vec{x}\|}). ~ \]
The gradient is viewed as a column vector. If the dot product above is viewed as matrix multiplication, then it would be written $\nabla{f}' \Delta{\vec{x}}$.
Linearization is the approximation
\[ ~ f(\vec{x}+\Delta{\vec{x}}) \approx f(\vec{x}) + \nabla{f}\cdot\Delta{\vec{x}}. ~ \]
The directional derivative of $f$ in the direction $\vec{v}$ is $\vec{v}\cdot\nabla{f}$, which can be seen as the derivative of the univariate function $(f\circ\vec{r})(t)$ where $\vec{r}(t) = p + t \vec{v}$.
For the function $z=f(x,y)$ the gradient points in the direction of steepest ascent. Ascent is seen in the 3d surface, the gradient is 2 dimensional.
For a function $f(\vec{x})$, a level curve is the set of values for which $f(\vec{x})=c$, $c$ being some constant. Plotted, this may give a curve or surface (in $n=2$ or $n=3$). The gradient at a point $\vec{x}$ with $f(\vec{x})=c$ will be orthogonal to the level curve $f=c$.
Partial derivatives are scalar functions, so have themselves partial derivatives. The notation $f_{xy}$ stands for the partial derivative in $y$ of the partial derivative of $f$ in $x$. Schwarz's theorem says the order of partial derivatives will not matter (e.g., $f_{xy} = f_{yx}$) provided the higher-order derivatives are continuous.
The chain rule applied to $(f\circ\vec{r})(t)$ says:
\[ ~ \frac{d(f\circ\vec{r})}{dt} = \nabla{f}(\vec{r}) \cdot \vec{r}'. ~ \]
For a function $F:R^n \rightarrow R^m$, the total derivative of $F$ is the linear operator $d_F$ satisfying:
\[ ~ \|F(\vec{x} + \vec{h})-F(\vec{x}) - d_F \vec{h}\| = \mathcal{o}(\|\vec{h}\|) ~ \]
For $F=\langle f_1, f_2, \dots, f_m\rangle$ the total derivative is the Jacobian, a $m \times n$ matrix of partial derivatives:
\[ ~ J_f = \left[ \begin{align}{} \frac{\partial f_1}{\partial x_1} &\quad \frac{\partial f_1}{\partial x_2} &\dots&\quad\frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1} &\quad \frac{\partial f_2}{\partial x_2} &\dots&\quad\frac{\partial f_2}{\partial x_n}\\ &&\vdots&\\ \frac{\partial f_m}{\partial x_1} &\quad \frac{\partial f_m}{\partial x_2} &\dots&\quad\frac{\partial f_m}{\partial x_n} \end{align} \right]. ~ \]
This can be viewed as being comprised of row vectors, each being the individual gradients; or as column vectors each being the vector of partial derivatives for a given variable.
The chain rule for $F:R^n \rightarrow R^m$ composed with $G:R^k \rightarrow R^n$ is:
\[ ~ d_{F\circ G}(a) = d_F(G(a)) d_G(a), ~ \]
That is the total derivative of $F$ at the point $G(a)$ times (matrix multiplication) the total derivative of $G$ at $a$. The dimensions work out as $d_F$ is $m\times n$ and $d_G$ is $n\times k$, so $d_(F\circ G)$ will be $m\times k$ and $F\circ{G}: R^k\rightarrow R^m$.
A scalar function $f:R^n \rightarrow R$ and a parameterized curve $\vec{r}:R\rightarrow R^n$ composes to yield a univariate function. The total derivative of $f\circ\vec{r}$ satisfies:
\[ ~ d_f(\vec{r}) d_\vec{r} = \nabla{f}(\vec{r}(t))' \vec{r}'(t) = \nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t), ~ \]
as above. (There is an identification of a $1\times 1$ matrix with a scalar in re-expressing as a dot product.)
Define the divergence of a vector-valued function $F:R^n \rightarrow R^n$ by:
\[ ~ \text{divergence}(F) = \frac{\partial{F_{x_1}}}{\partial{x_1}} + \frac{\partial{F_{x_2}}}{\partial{x_2}} + \cdots \frac{\partial{F_{x_n}}}{\partial{x_n}}. ~ \]
The divergence is a scalar function. For a vector field $F$, it measures the microscopic flow out of a region.
A vector field whose divergence is identically $0$ is called incompressible.
Define the curl of a two-dimensional vector field, $F:R^2 \rightarrow R^2$, by:
\[ ~ \text{curl}(F) = \frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}}. ~ \]
The curl for $n=2$ is a scalar function.
For $n=3$ define the curl of $F:R^3 \rightarrow R^3$ to be the vector field:
\[ ~ \text{curl}(F) = \langle \ \frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}}, \frac{\partial{F_x}}{\partial{z}} - \frac{\partial{F_z}}{\partial{x}}, \frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}} \rangle. ~ \]
The curl measures the circulation in a vector field. In dimension $n=3$ it points in the direction of the normal of the plane of maximum circulation with direction given by the right-hand rule.
A vector field whose curl is identically of magnitude $0$ is called irrotational.
The $\nabla$ operator is the formal vector $~ \nabla = \langle \frac{\partial}{\partial{x}}, \frac{\partial}{\partial{y}}, \frac{\partial}{\partial{z}} \rangle. ~$
The gradient is then scalar "multiplication" on the left: $\nabla{f}$.
The divergence is the dot product on the left: $\nabla\cdot{F}$.
The curl is the the cross product on the left: $\nabla\times{F}$.
These operations satisfy two vanishing properties:
The curl of a gradient is the zero vector: $\nabla\times\nabla{f}=\vec{0}$
The divergence of a curl is $0$: $\nabla\cdot(\nabla\times F)=0$
Helmholtz decomposition theorem says a vector field ($n=3$) which vanishes rapidly enough can be expressed in terms of $F = -\nabla\phi + \nabla\times{A}$. The left term will be irrotational (no curl) and the right term will be incompressible (no divergence).
The definite integral, $\int_a^b f(x) dx$, for a bounded univariate function is defined in terms Riemann sums, $\lim \sum f(c_i)\Delta{x_i}$ as the maximum partition size goes to $0$. Similarly the integral of a bounded scalar function $f:R^n \rightarrow R$ over a box-like region $[a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_n,b_n]$ can be defined in terms of a limit of Riemann sums. A Riemann integrable function is one for which the upper and lower Riemann sums agree in the limit. A characterization of a Riemann integrable function is that the set of discontinuities has measure $0$.
If $f$ and the partial functions ($x \rightarrow f(x,y)$ and $y \rightarrow f(x,y)$) are Riemann integrable, then Fubini's theorem allows the definite integral to be performed iteratively:
\[ ~ \iint_{R\times S}fdV = \int_R \left(\int_S f(x,y) dy\right) dx = \int_S \left(\int_R f(x,y) dx\right) dy. ~ \]
The integral satisfies linearity and monotonicity properties that follow from the definitions:
For integrable $f$ and $g$ and constants $a$ and $b$:
\[ ~ \iint_R (af(x) + bg(x))dV = a\iint_R f(x)dV + b\iint_R g(x) dV. ~ \]
If $R$ and $R'$ are disjoint rectangular regions (possibly sharing a boundary), then the integral over the union is defined by linearity:
\[ ~ \iint_{R \cup R'} f(x) dV = \iint_R f(x)dV + \iint_{R'} f(x) dV. ~ \]
As $f$ is bounded, let $m \leq f(x) \leq M$ for all $x$ in $R$. Then
\[ ~ m V(R) \leq \iint_R f(x) dV \leq MV(R). ~ \]
If $f$ and $g$ are integrable and$f(x) \leq g(x)$, then the integrals have the same property, namely $\iint_R f dV \leq \iint_R gdV$.
If $S \subset R$, both closed rectangles, then if $f$ is integrable over $R$ it will be also over $S$ and, when $f\geq 0$, $\iint_S f dV \leq \iint_R fdV$.
If $f$ is bounded and integrable, then $|\iint_R fdV| \leq \iint_R |f| dV$.
In two dimensions, we have the following interpretations:
\[ ~ \begin{align} \iint_R dA &= \text{area of } R\\ \iint_R \rho dA &= \text{mass with constant density }\rho\\ \iint_R \rho(x,y) dA &= \text{mass of region with density }\rho\\ (1/\text{area})\iint_R x \rho(x,y)dA &= \text{centroid of region in } x \text{ direction}\\ (1/\text{area})\iint_R y \rho(x,y)dA &= \text{centroid of region in } y \text{ direction} \end{align} ~ \]
In three dimensions, we have the following interpretations:
\[ ~ \begin{align} \iint_VdV &= \text{volume of } V\\ \iint_V \rho dV &= \text{mass with constant density }\rho\\ \iint_V \rho(x,y) dV &= \text{mass of volume with density }\rho\\ (1/\text{volume})\iint_V x \rho(x,y)dV &= \text{centroid of volume in } x \text{ direction}\\ (1/\text{volume})\iint_V y \rho(x,y)dV &= \text{centroid of volume in } y \text{ direction}\\ (1/\text{volume})\iint_V z \rho(x,y)dV &= \text{centroid of volume in } z \text{ direction} \end{align} ~ \]
To compute integrals over non-box-like regions, Fubini's theorem may be utilized. Alternatively, a transformation of variables
For a parameterized curve, $\vec{r}(t)$, the line integral of a scalar function between $a \leq t \leq b$ is defined by: $\int_a^b f(\vec{r}(t)) \| \vec{r}'(t)\| dt$. For a path parameterized by arc-length, the integral is expressed by $\int_C f(\vec{r}(s)) ds$ or simply $\int_C f ds$, as the norm is $1$ and $C$ expresses the path.
A Jordan curve in two dimensions is a non-intersecting continuous loop in the plane. The Jordan curve theorem states that such a curve divides the plane into a bounded and unbounded region. The curve is positively parameterized if the the bounded region is kept on the left. A line integral over a Jordan curve is denoted $\oint_C f ds$.
Some interpretations: $\int_a^b \| \vec{r}'(t)\| dt$ computes the arc-length. If the path represents a wire with density $\rho(\vec{x})$ then $\int_a^b \rho(\vec{r}(t)) \|\vec{r}'(t)\| dt$ computes the mass of the wire.
The line integral is also defined for a vector field $F:R^n \rightarrow R^n$ through $\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt$. When parameterized by arc length, this becomes $\int_C F(\vec{r}(s)) \cdot \hat{T} ds$ or more simply $\int_C F\cdot\hat{T}ds$. In dimension $n=2$ if $\hat{N}$ is the normal, then this line integral (the flow) is also of interest $\int_a^b F(\vec{r}(t)) \cdot \hat{N} dt$ (this is also expressed by $\int_C F\cdot\hat{N} ds$).
When $F$ is a force field, then the interpretation of $\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt$ is the amount of work to move an object from $\vec{r}(a)$ to $\vec{r}(b)$. (Work measures force applied times distance moved.)
A conservative force is a force field within an open region $R$ with the property that the total work done in moving a particle between two points is independent of the path taken. (Similarly, integrals over Jordan curves are zero.)
The gradient theorem or fundamental theorem of line integrals states if $\phi$ is a scalar function then the vector field $\nabla{\phi}$ (if continuous in $R$) is a conservative field. That is if $q$ and $p$ are points, $C$ any curve in $R$, and $\vec{r}$ a parameterization of $C$ over $[a,b]$ that $\phi(p) - \phi(q) = \int_a^b \nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t) dt$.
If $\phi$ is a scalar function producing a field $\nabla{\phi}$ then in dimensions $2$ and $3$ the curl of $\nabla{\phi}$ is zero when the functions involved are continuous. Conversely, if the curl of a force field, $F$, is zero and the derivatives are continuous in a simply connected domain, then there exists a scalar potential function, $\phi$, with $F = -\nabla{\phi}$.
In dimension $2$, if $F$ describes a flow field the integral $\int_C F \cdot\hat{N}ds$ is interpreted as the flow across the curve $C$; when $C$ is a closed curve $\oint_C F\cdot\hat{N}ds$ is interpreted as the flow out of the region, when $C$ is positively parameterized.
Green's theorem states if $C$ is a positively oriented Jordan curve in the plane bounding a region $D$ and $F$ is a vector field $F:R^2 \rightarrow R^2$ then $\oint_C F\cdot\hat{T}ds = \iint_D \text{curl}(F) dA$.
Green's theorem can be re-expressed in flow form: $\oint_C F\cdot\hat{N}ds=\iint_D\text{divergence}(F)dA$.
For $F=\langle -y,x\rangle$, Green's theorem says the area of $D$ is given by $(1/2)\oint_C F\cdot\vec{r}' dt$. Similarly, if $F=\langle 0,x\rangle$ or $F=\langle -y,0\rangle$ then the area is given by $\oint_C F\cdot\vec{r}'dt$. The above follows as $\text{curl}(F)$ is $2$ or $1$. Similar formulas can be given to compute the centroids, by identifying a vector field with $\text{curl}(F) = x$ or $y$.
A surface in $3$ dimensions can be described by a scalar function $z=f(x,y)$, a parameterization $F:R^2 \rightarrow R^3$ or as a level curve of a scalar function $f(x,y,z)$. The second case, covers the first through the parameterization $(x,y) \rightarrow (x,y,f(x,y)$. For a parameterization of a surface, $\Phi(u,v) = \langle \Phi_x, \Phi_y, \Phi_z\rangle$, let $\partial{\Phi}/\partial{u}$ be the $3$-d vector $\langle \partial{\Phi_x}/\partial{u}, \partial{\Phi_y}/\partial{u}, \partial{\Phi_z}/\partial{u}\rangle$, similarly define $\partial{\Phi}/\partial{v}$. As vectors, these lie in the tangent plane to the surface and this plane has normal vector $\vec{N}=\partial{\Phi}/\partial{u}\times\partial{\Phi}/\partial{v}$. For a closed surface, the parametrization is positive if $\vec{N}$ is an outward pointing normal. Let the surface element be defined by $\|\vec{N}\|$.
The surface integral of a scalar function $f:R^3 \rightarrow R$ for a parameterization $\Phi:R \rightarrow S$ is defined by $~ \iint_R f(\Phi(u,v)) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv ~$
If $F$ is a vector field, the surface integral may be defined as a flow across the boundary through
\[ ~ \iint_R F(\Phi(u,v)) \cdot \vec{N} du dv = \iint_R (F \cdot \hat{N}) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv = \iint_S (F\cdot\hat{N})dS ~ \]
Stokes' theorem states that in dimension $3$ if $S$ is a smooth surface with boundary $C$oriented so the right-hand rule gives the choice of normal for $S$, and $F$ is a vector field with continuous partial derivatives then
\[ ~ \iint_S (\nabla\times{F}) \cdot \hat{N} dS = \oint_C F ds. ~ \]
Stokes' theorem has the same formulation as Green's theorem in dimension $2$, where the surface integral is just the $2$-dimensional integral.
Stokes' theorem is used to show a vector field $F$ with zero curl is conservative if $F$ is continuous in a simply connected region.
Stokes' theorem is used in Physics, for example, to relate the differential and integral forms of 2 of Maxwell's equations.
The divergence theorem states if $V$ is a compact volume in $R^3$ with piecewise smooth boundary $S=\partial{V}$ and $F$ is a vector field with continuous partial derivatives then
\[ ~ \iint_V (\nabla\cdot{F})dV = \oint_S (F\cdot\hat{N})dS. ~ \]
The divergence theorem is available for other dimensions. In the $n=2$ case, it is the alternate (flow) form of Green's theorem.
The divergence theorem is used in Physics to express physical laws in either integral or differential form.